We introduce the Illustris Project, a series of large-scale hydrodynamical simulations of galaxy formation. The highest resolution simulation, Illustris-1, covers a volume of $(106.5,{rm Mpc})^3$, has a dark mass resolution of ${6.26 times 10^{6},{rm M}_odot}$, and an initial baryonic matter mass resolution of ${1.26 times 10^{6},{rm M}_odot}$. At $z=0$ gravitational forces are softened on scales of $710,{rm pc}$, and the smallest hydrodynamical gas cells have an extent of $48,{rm pc}$. We follow the dynamical evolution of $2times 1820^3$ resolution elements and in addition passively evolve $1820^3$ Monte Carlo tracer particles reaching a total particle count of more than $18$ billion. The galaxy formation model includes: primordial and metal-line cooling with self-shielding corrections, stellar evolution, stellar feedback, gas recycling, chemical enrichment, supermassive black hole growth, and feedback from active galactic nuclei. At $z=0$ our simulation volume contains about $40,000$ well-resolved galaxies covering a diverse range of morphologies and colours including early-type, late-type and irregular galaxies. The simulation reproduces reasonably well the cosmic star formation rate density, the galaxy luminosity function, and baryon conversion efficiency at $z=0$. It also qualitatively captures the impact of galaxy environment on the red fractions of galaxies. The internal velocity structure of selected well-resolved disk galaxies obeys the stellar and baryonic Tully-Fisher relation together with flat circular velocity curves. In the well-resolved regime the simulation reproduces the observed mix of early-type and late-type galaxies. Our model predicts a halo mass dependent impact of baryonic effects on the halo mass function and the masses of haloes caused by feedback from supernova and active galactic nuclei.
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